Some asymptotics for Sobolev orthogonal polynomials involving Gegenbauer weights

Abstract

We consider the Sobolev inner product< f.g > = integral(1)(-1) f(x)g(x)(1 - x(2))(alpha-1/2) dx + integral f'(x)g'(x)d psi(x), alpha > -1/2,where d(psi) is a measure involving a Gegenbauer weight and with mass points outside the interval (-1, 1). We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product. We obtain the asymptotics of the largest zeros of these polynomials via a Mehler-Heine type formula. These results are illustrated with some numerical experiments. (C) 2010 Elsevier B.V. All rights reserved.